The distance scale to cosmic gamma-ray bursts (GRB's) is still uncertain by many orders of magnitude; however, one viable scenario places GRB's at cosmological distances, thereby permitting them to be used as tracers of the cosmological expansion over a significant range of redshifts zeta. Also, several recent measurements of the Hubble constant H(sub 0) appearing in the referred literature report values of 70-80 km/s /Mpc. Although there is significant debate regarding these measurements, we proceed here under the assumption that they are evidence of a large value for H(sub 0). This is done in order to investigate the additional constraints on cosmological models that can be obtained under this hypothesis when combined with the age of the universe and the brightness distribution of cosmological gamma-ray bursts. We show that the range of cosmological models that can be consistent with the GRB brightness distribution, a Hubble constant of 70-80 km/s/Mpc, and a minimum age of the universe of 13-15 Gyr is constrained significantly, largely independent of a wide range of assumptions regarding the evolutionary nature of the burst population. Low-density, Lambda greater than 0 cosmological models with deceleration parameter in the range -1 less than q(sub 0) less than 0 and density parameter sigma(sub 0) in the range approximately equals 0.10-0.25(Omega(sub 0) approximately equals 0.2-0.5) are strongly favored.